Editing Parseval's identity (section)

== Generalization of the Pythagorean theorem ==

The [[Identity (mathematics)|identity]] is related to the [[Pythagorean theorem]] in the more general setting of a [[Separable (topology)|separable]] [[Hilbert space]] as follows.  Suppose that <math>H</math> is a Hilbert space with [[inner product]] <math>\langle \,\cdot\,, \,\cdot\, \rangle.</math>  Let <math>\left(e_n\right)</math> be an [[orthonormal basis]] of <math>H</math>; i.e., the [[linear span]] of the <math>e_n</math> is [[Dense set|dense]] in <math>H,</math> and the <math>e_n</math> are mutually orthonormal:
:<math>\langle e_m, e_n\rangle = \begin{cases}
1 & \mbox{if}~ m = n \\
0 & \mbox{if}~ m \neq n.
\end{cases}</math>

Then Parseval's identity asserts that for every <math>x \in H,</math>
<math display="block">\sum_n \left|\left\langle x, e_n \right\rangle\right|^2 = \|x\|^2.</math>

This is directly analogous to the [[Pythagorean theorem]], which asserts that the sum of the squares of the components of a vector in an orthonormal basis is equal to the squared length of the vector.  One can recover the Fourier series version of Parseval's identity by letting <math>H</math> be the Hilbert space <math>L^2[-\pi, \pi],</math> and setting <math>e_n = e^{i n x}</math> for <math>n \in \Z.</math>

More generally, Parseval's identity holds for arbitrary [[Hilbert space|Hilbert spaces]], not necessarily separable. When the Hilbert space is not separable any orthonormal basis is uncountable and we need to generalize the concept of a series to an unconditional sum as follows: let <math>\{e_r\}_{r\in \Gamma}</math> an orthonormal basis of a Hilbert space (where <math>\Gamma</math> have arbitrary cardinality), then we say that <math display="inline">\sum_{r\in \Gamma} a_r e_r</math> converges unconditionally if for every <math>\epsilon>0</math> there exists a finite subset <math>A\subset \Gamma</math> such that
<math display="block">
\left\| \sum_{r\in B}a_re_r-\sum_{r\in C}a_r e_r\right\|<\epsilon
</math>
for any pair of finite subsets <math>B,C\subset\Gamma</math> that contains <math>A</math> (that is, such that <math>A\subset B\cap C</math>). Note that in this case we are using a [[Net (mathematics)|net]] to define the unconditional sum.